2k  =  [y1 + (k-1) v1]   [y2 + (k-1) v2]


With all parameters set to 1,
these coefficients \(\alpha _{2k-1} = \alpha _{2k} = k^2\) 
correspond to the continued fraction
[1]} [2]}
for the median Genocchi numbers;
so it is natural to seek a combinatorial model
that is enumerated by the median Genocchi numbers.
Many such models are known.
In this paper we shall focus on a class of permutations of \([2n]\) 
called D-permutations [3]}, [4]},
which are defined by imposing some constraints
concerning the parity (even/odd) of excedances and anti-excedances.
Recall that an index \(i\)  in a permutation \(\sigma \) 
is called an excedance in case \(i < \sigma (i)\) ,
an anti-excedance in case \(i > \sigma (i)\) ,
and a fixed point in case \(i = \sigma (i)\) .
A permutation is called a D-permutation
in case it contains no even excedances and no odd anti-excedances.
Equivalently, a permutation is a D-permutation in case
\(2k-1 \le \sigma (2k-1)\)  and \(2k \ge \sigma (2k)\)  for all \(k\) .
Let us say also that a permutation is an
e-semiderangement (resp. o-semiderangement)
in case in contains no even (resp. odd) fixed points;
it is a derangement in case it contains no fixed points at all.
A D-permutation that is also an
e-semiderangement (resp. o-semiderangement, derangement)
will be called a D-e-semiderangement
(resp. D-o-semiderangement, D-derangement).In the past, D-o-semiderangements have been called
Genocchi permutations
[5]}, [6]}, [7]}
or excedance-alternating permutations
[8]};
D-e-semiderangements [9]}
have been called Dumont permutations [3]}, [4]}, [12]}
or Dumont permutations of the second kind
[13]}, [14]};
D-derangements have been called
Dumont derangements [3]}, [4]}, [12]}.
D-permutations that are not semiderangements were apparently
first considered in the recent work of
Lazar and Wachs [3]}, [4]}.
Note also that the involution of \({\mathfrak {S}}_{2n}\)  defined by
\(\sigma \mapsto R \circ \sigma \circ R\) ,
where \(R(i) = 2n+1-i\)  is the reversal map,
takes D-permutations into D-permutations
and interchanges e-semiderangements with o-semiderangements;
it therefore yields a bijection between
D-e-semiderangements and D-o-semiderangements.
Therefore, any result about one of the two types of D-semiderangements
can be expressed equivalently in terms of the other.

A D-permutation that contains exactly one cycle is called a D-cycle. Notice that a D-cycle is also a D-derangement.
Let \({\mathfrak {D}}_{2n}\) 
(resp. \({\mathfrak {D}}^{\rm e}_{2n}, {\mathfrak {D}}^{\rm o}_{2n}, {\mathfrak {D}}^{\rm eo}_{2n}, {\mathfrak {DC}}_{2n}\) )
denote the set of all D-permutations
(resp. D-e-semiderangements, D-o-semiderangements, D-derangements, D-cycles) of \([2n]\) .
For instance,

D2  =  { 12 ,  21eo }
